Bound States at Threshold Resulting from Coulomb Repulsion

Abstract

The eigenvalue absorption for a many–particle Hamiltonian depending on a
parameter is analyzed in the framework of
non–relativistic quantum mechanics. The long–range part of pair potentials is
assumed to be pure Coulomb and no restriction on the particle statistics is
imposed.
It is proved that if the lowest
dissociation threshold corresponds to the decay into two likewise non–zero
charged clusters then the bound state,
which approaches the threshold, does not spread and eventually
becomes the bound state at threshold. The obtained results have
applications in atomic and nuclear physics. In particular, we prove that an
atomic
ion with the critical charge Zcr and Ne electrons
has a bound state at threshold given that Zcr∈(Ne−2,Ne−1),
whereby the electrons are treated as fermions and the mass of the nucleus
is finite.

I Introduction

In Refs. 1, ; 2, it was proved that a critically bound N–body system, where none
of the subsystems has bound states with E≤0 and
particle pairs have no zero energy resonances,
has a square integrable state at zero energy. The
condition on the absence of 2–body zero energy resonances was shown to be
essential in the three–body case 1 (). Here we consider the N–particle
system,
where particles can be charged and apart from short–range pair–interactions
may also interact
via Coulomb attraction/repulsion. The formation of bound states at threshold in
the two–particle case when the particles Coulomb repel each other is
well–studied 3 (); gest (). In the three–particle case there is a well–known proof
ostenhof () that a two–electron ion with an infinitely heavy nucleus has a bound state at
threshold, when the nuclear charge becomes critical.

Our aim here is to
investigate the general many–particle case. Here we generalize the result in
Ref. ostenhof, to the case of many electron ions with Fermi statistics
and finite nuclear mass. In the proofs we shall use the bounds on Green’s
functions from Ref. 3, as well as the technique of spreading sequences from
Ref. 1, ,
that is we prove the eigenvalue absorption by demonstrating that the wave
functions corresponding to bound states do not spread, c.f. Theorem 1 in
Ref. 1, . A different approach based on the calculus of variations was recently developed in Ref. frank, , where the authors
give an alternative proof to the result in Ref. ostenhof, . The authors in Ref. frank, indicate that their approach could be
generalized to the many–particle case. In the present paper as well as in Refs. ostenhof, ; frank, one uses essentially the same idea, namely, one uses the fact that
the weak limit of ground state wave functions is a solution to the Schrödinger equation at the threshold. The hardest part is to prove that the weak limit is not
identically zero. Our approach differs from the ones in Refs. ostenhof, ; frank, in that we use the upper bounds on the two–particle Green’s functions3 ().

The paper is organized as follows. In Sec. II we introduce notations,
formulate the main theorem and prove a number of technical lemmas.
In Sec. III we derive an upper bound on the Green’s function, which is
used in Sec. IV for the proof of Theorem 1.
In Sec. V we discuss two main applications
of Theorem 1 concerning the stability diagram of three Coulomb
charges (Theorem 2 in Sec. V.1) and negative atomic
ions
(Theorem 3 in Sec. V.2). In Appendix A we
derive various criteria for non–spreading sequences.

Let us mention physical applications. The effect when a
size of a bound system increases near the threshold and by far exceeds the
scales set by attractive parts of potentials was discovered in
neutron halos, helium dimer, Efimov states, for
discussion see Refs. fedorov, ; zhukov, ; efimov, ; hansen0, . Here we
demonstrate that in a many–particle system similarly to the two–body case
3 (); gest () a Coulomb repulsion between possible decay
products blocks the spreading of bound states and forces an L2 bound state at
threshold. In
nuclear physics, this, in particular, explains why contrary to neutron halos no
proton halos are found hansen ().

Ii Formulation of the Main Theorem

We consider the N–particle Hamiltonian (N≥3)

H(λ)=H0+V(λ),

(1)

V(λ):=∑1≤i<j≤NVij(λ)≡∑1≤i<j≤N[Uij(λ;ri−rj)+qi(λ)qj(λ)|ri−rj|],

(2)

where λ∈R is a parameter, H0 is the kinetic energy
operator with the center of mass
removed, ri∈R3 are particles’ position vectors and qi(λ)∈R
denote the particles’ charges depending on λ. We shall assume that
Uij(λ;r)∈L2(R3)+L∞∞(R3)
for each given
λ. Here L∞∞(Rn)
denotes the space of bounded Borel functions vanishing at infinity.
We shall also take particle spins into account, though we shall consider only
spin–independent Hamiltonians.
The Hamiltonian acts in
L2(R3N−3)⊕L2(R3N−3)⊕⋯⊕L2(R3N−3)≡L2(R3N−3;Cns),
where the direct sum has ns=(2s1+1)(2s2+1)…(2sN+1)
summands and si denotes the spin of particle i.
Similar notation for the Hilbert space can be found in Refs. quotesimon, ; quotethaller, .
By Kato’s theorem reed (); teschl ()H(λ) is self–adjoint on
D(H0)=H2(R3N−3;Cns)⊂L2(R3N−3;Cns), where
H2(R3N−3;Cns)≡H2(R3N−3)⊕⋯⊕H2(R3N−3) and
H2(R3N−3) denotes the
corresponding Sobolev space teschl (); liebloss (). A function f∈L2(R3N−3;Cns) depends explicitly on the arguments as
f(x,σ1,…,σN), where x∈R3N−3 and σi∈{si2,si2−1,…,−si2} are the spin
variables.

We treat the particles with integer spins as bosons and particles with
half–integer spin as fermions.
P denotes the orthogonal projection operator on the
subspace of functions, which are symmetric with respect to
the interchange of bosons and antisymmetric with respect to the interchange of
fermions. We denote the
bottom of the continuous spectrum by

Ethr(λ):=infσessH(λ)P.

(3)

We shall use the function ηα:Rn→R, which determines the asymptotic behavior at
infinity

ηα(r):=χ{r||r|≤1}+χ{r||r|>1}|r|α,

(4)

where r∈Rn,α∈R+ and χA always denotes
the
characteristic function of the set A.
Note that ηα(r) is continuous and ηα1ηα2=ηα1α2. We make the following assumptions

H(λ) is defined for an infinite sequence of parameter values
λ1,λ2,… and λcr, where limn→∞λn=λcr. For all λn there is E(λn)∈R,ψn∈D(H0)
such that H(λn)ψn=E(λn)ψn, where ∥ψn∥=1,
Pψn=ψn
and E(λn)<Ethr(λn). Besides, limn→∞E(λn)=limn→∞Ethr(λn)=Ethr(λcr).

supλ=λn,λcr|Uij(λ;y)|≤~U(y) and supλ=λn,λcr|qi(λ)qj(λ)|≤q0, where ~U(y) is such that
ηδ(y)~U(y)∈L2(R3)+L∞∞(R3) and δ∈(3/2,2), q0∈(0,∞)
are fixed constants.
Additionally, for all f∈C∞0(R3N−3).

Let a=1,2,…,(2N−1−1) label all the distinct ways ims () of partitioning
particles into two non–empty clusters Ca1 and
Ca2. We define the Jacobi intercluster coordinates for the
clusters Ca1,2 as
xa,1i and xa,2j respectively, where i=1,2,…,(#Ca1−1) and
j=1,2,…,(#Ca2−1) (the symbol # denotes the
number
of particles in the corresponding cluster).
By xa we denote the full set of intercluster
coordinates and we set

|xa|=#Ca1−1∑i=1|xa,1i|+#Ca2−1∑j=1|xa,2j|.

(5)

Ra points from the center
of mass
of Ca1 to the center of mass of Ca2. The full
set of Jacobi coordinates
is (xa,Ra)∈R3N−3.

We denote the sum of interaction cross terms between the clusters by

Ia(λ):=∑i∈Ca1j∈Ca2Vij(λ).

(6)

The product of net charges of the clusters is defined as

Qa(λ):=∑i∈Ca1∑j∈Ca2qi(λ)qj(λ).

(7)

The projection operators on the proper symmetry subspace for the particles
within clusters C(a)1 and C(a)2 are
P(a)1 and P(a)2 respectively. Namely,
P(a)i projects on a subspace of functions,
which are antisymmetric with respect to the
interchange of fermions in C(a)i and symmetric with respect to
the interchange of bosons in C(a)i (i=1,2).
Naturally,
and [P(a)1,P(a)2]=0. We also
define P(a):=P(a)1P(a)2. The
Hamiltonian (1) can be decomposed in the following way

H(λ)=H(a)thr(λ)−ℏ22μaΔRa+Ia(λ),

(8)

where H(a)thr(λ) is the Hamiltonian of the clusters’ intrinsic
motion
and μa denotes the reduced mass derived from clusters’ total masses. From
now on without loss of generality we set ℏ2/(2μa)=1.

It is convenient to treat the Hilbert space as the tensor product
L2(R3N−3;Cns)=L2(R3N−6;Cns)⊗L2(R3), where the
first term in the product
corresponds to the space associated with xa coordinates and spin variables, while the second one
refers to the space
associated with the Ra coordinate. In such case the operator H(a)thr
has the form
H(a)thr=Hathr⊗1, where Hathr is the restriction
of H(a)thr to L2(R3N−6;Cns). The
coordinate Ra is unaffected by permutations of particles
within the clusters C(a)1 or
C(a)2. Therefore, P(a)=Pa⊗1, where Pa denotes the restriction of
P(a) to the space associated with xa coordinates and spin variables.

The set of assumptions is continued as follows.

For λ=λn,λcr and a=1,…,N one has
infσ(Hathr(λ)Pa)=Ethr(λ). There is
|Δϵ|>0 such that the following
inequalities hold for λ=λn,λcr

(9)

[Hathr(λ)−Ethr(λ)]Pa≥|Δϵ|Pa(a=N+1,…,2N−1−1).

(10)

The requirement R3 says that the bottom of the continuous spectrum of
H(λ) is
set by the decomposition into those two clusters that correspond to any of the
decompositions a=1,…,N. Inequality (9) introduces
a gap
between the ground state energy of the two clusters and other excited states.
For
a=1,…,N and λ=λn,λcr we define the
projection operator
acting on L2(R3N−6;Cns)

Pathr(λ)=Pa[Ethr(λ),Ethr(λ)+|Δϵ|],

(11)

where {PaΩ} are spectral projections of
Hathr(λ)Pa. Note that by R3 the projection operators Pathr(λn), Pathr(λcr)
have a finite dimensional range.

The last assumption introduces the uniform control over the fall off of clusters’ wave
functions

There are constants A,β>0 such that

∥∥eβ|xa|Pathr(λ)∥∥≤A

(12)

for λ=λn,λcr and a=1,2,…,N.

Due to R3 there must exist orthonormal φai(λ)∈D(−Δ)⊂L2(R3N−6;Cns) for i=1,2,…,na(λ) such that

Pathr(λ)=na(λ)∑i=1Eai(λ)φai(λ)(⋅,φai(λ))for a=1,…,N and λ=λn,λcr,

(13)

where Eai(λ)∈[Ethr(λ),Ethr(λ)+|Δϵ|]. Note that Hathr(λn)φai(λn)=Eai(λn)φai(λn), therefore,
∥−Δφai(λn)∥ is uniformly bounded, c. f. Lemma 1 in
Ref. 1, . Applying Lemma 1 below and using R4 we conclude that
there exists an integer ω such that na(λcr),na(λn)≤ω.

Lemma 1.

Suppose that the orthonormal set of function ϕ1,…,ϕN∈D(−Δ)⊂L2(Rd;Cns) is
such that
∥−Δϕi∥≤T and ∥eβ|x|ϕi∥≤A for i=1,…,N, where T,A,β>0 are constants. If d≥3 then

N≤Cd(2T)d/2|ln2A|d(2β)dns,

(14)

where Cd is the Lieb’s constant in the Cwikel–Lieb–Rosenbljum bound.

Proof.

From ∥eβ|x|ϕi∥≤A it follows that

(ϕi,χ{x||x|≤R}ϕi)≥12,

(15)

where we set R:=(ln2A)/(2β). Hence,

(16)

By the min–max principle N does not exceed the number of negative
energy bound states of the operator in square brackets in (16). This
number, in turn, is equal to the number of negative energy bound states of the
operator in square brackets considered in L2(Rd) times ns due
to the spin degeneracy. Now
(14) follows from the Cwikel–Lieb–Rosebljum bound reed (); cwikel ().
∎

Now we can formulate the main theorem.

Theorem 1.

Suppose that H(λ) satisfies R1−R4 and

Q0:=infa=1,…,Ninfλ=λn,λcrQa(λ)>0.

(17)

Then the sequence ψn does not spread and there exists ψcr∈D(H0)⊂L2(R3N−3;Cns) such that
H(λcr)ψcr=Ethr(λcr)ψcr, where
∥ψcr∥=1 and ψcr=Pψcr.

Let us remark that the term spreading was defined in Ref. 1, for sequences in
L2(Rd). We shall say that a sequence
fn∈L2(Rd;Cns) spreads if fn(x,σ1,…,σN) spreads for all possible fixed values of the spin
variables.
We postpone the proof of Theorem 1 to Sec. IV.

Together with the upper bound on the Green’s function derived in the next
section the following lemma is the
key ingredient in the proof of Theorem 1.

Lemma 2.

There is Θa(x)∈L2(R3N−3)+L∞∞(R3N−3) independent of λ such that

∣∣e−β|xa|ηδ(Ra)[Ia(λ)−Qa(λ)η−1(Ra)]∣∣≤Θa(x)

(18)

for λ=λn,λcr defined in R1, δ defined in R2 and
β defined in R4.

Proof.

The statement of the lemma is based on the following inequality, which can be
checked directly. For all s,s′∈R3

∣∣∣χ{s,s′||s−s′|≥1}|s−s′|−η−1(s)∣∣∣≤2η2(s′)η−2(s).

(19)

For fixed s′ the term on the lhs of (19) falls off like |s|−2. We
write

where ca,1i, ca,2i are numerical coefficients depending on masses.
It is easy to see that the coefficient in front of Ra is always 1 by fixing
|xa| and
taking |Ra|≫1. Therefore, by (19) we have

∣∣∣χ{x||ri−rj|≥1}|ri−rj|−η−1(Ra)∣∣∣≤c0η2(|xa|)η−2(Ra),

(22)

where c0>0 is some constant. Substituting (22) into (20)
we conclude that the inequality (18) would be true if we set Θa=Θa1+Θa2, where

Θa1(x):=e−β|xa|ηδ(Ra)∑i∈Ca1j∈Ca2[~Uij+q0|ri−rj|χ{x||ri−rj|≤1}],

(23)

Θa2(x):=c0N(N−1)q0e−β|xa|η2(|xa|)ηδ−2(Ra).

(24)

Using R2 it is easy to see that Θa1∈L2(R3N−3)+L∞∞(R3N−3). Because δ<2 we have Θa2∈L∞∞(R3N−3).
∎

Iii Upper Bound on the Two Particle Green’s Function

Consider the following integral operator on L2(R3)

Gck(A)=[−Δ+Aη−1(r)+k2]−1,

(25)

for A,k>0, whose integral kernel we denote as Gck(A;r,r′) (the
superscript “c” refers to
“Coulomb”). Note that Gck(A;r,r′)≤Gc~k(~A;r,r′) away from r=r′ if either
~A≤A or ~k≤k, c. f. Lemma 1 in Ref. 3, . The following
Lemma uses the upper bound on a two particle Green’s
function from Ref. 3, .

Lemma 3.

There is b(A)>0 such that for all A>0, n>0

supk>0∥∥Gck(A)χ{r||r|≤n}∥∥≤b(A)n,

(26)

where the norm on the lhs is the operator norm.

Proof.

The operator Gck(A) is an integral operator with a positive kernel
lpestim () and, hence, it suffices to consider (26) for n>1.
For a shorter notation we denote χn:=χ{r||r|≤n}.
Obviously

∥Gck(A)χn∥≤∥χ4nGck(A)χn∥+∥(1−χ4n)Gck(A)χn∥

≤∥χ4nGck(A)χ4n∥+∥(1−χ4n)Gck(A)χn∥,

(27)

where the last inequality follows from Gck(A) being an integral
operator
with a positive kernel 3 (); lpestim (). We shall derive the following estimates
∥χ4nGck(A)χ4n∥=O(n) and
∥(1−χ4n)Gck(A)χn∥=o(n) for n→∞,
from which
the
the statement of the Lemma follows. The first term on the rhs of (27) is
the norm of the self–adjoint operator, which can be estimated as follows

∥χ4nGck(A)χ4n∥=sup∥f∥=1(χ4nf,Gck(A)χ4nf)≤

(28)

where we have used the inequality (B+ε)−1≤(C+ε)−1 for non–negative self–adjoint operators B≥C≥0 and ε>0 (see, for example Ref. glimmjaffe, , Proposition
A.2.5 on page 131). Thus ∥χ4nGck(A)χ4n∥=O(n) as claimed.

Let us now consider the second term on the rhs of (27). We
shall need the bound on the Green’s function from Ref. 3, . Let ~Gk(a;r,r′) denote the integral kernel of the following operator on L2(R3)

~Gk(a)=[−Δ+(a24|r|−1+a4|r|−3/2)χ{r||r|≥1}+k2]−1.

(29)

Lets us set a equal to the positive root of the equation a(a+1)=4A. Then
we get

Aη−1(r)≥(a24|r|−1+a4|r|−3/2)χ{r||r|≥1},

(30)

which means that Gck(A;r,r′)≤~Gk(a;r,r′) pointwise for
all r≠r′, see Ref. 3, . The upper bound on ~Gk(a;r,r′) from
Ref. 3, (Eqs.(42)–(43) and Eqs. (39)–(40) in Ref. 3, ) reads

where we have set ~R0=2n and ~a=a/2. It is straightforward
to check that this choice of ~R0,~a indeed satisfies
(32)–(33). Taking into account that Gck(A;r,r′)≤~Gk(a;r,r′) we finally get from (34) the required bound

Lemma 4.

Proof.

For an arbitrary f∈L2(R3) we have

∥∥Gck(A)η−αf∥∥=limN→∞∥∥N∑n=1Gck(A)η−α(χn−χn−1)f∥∥

≤limN→∞N∑n=1∥∥Gck(A)χnη−α(χn−χn−1)2f∥∥,

(38)

where we have used (χn−χn−1)2=(χn−χn−1)
and χn(χn−χn−1)=(χn−χn−1).
For the operator norms we have ∥η−αχ1∥=1 and ∥η−α(χn−χn−1)∥=(n−1)−α for
n≥2. Substituting these into (38) and using Lemma 3 we
rewrite (38) as

∥Gck(A)η−αf∥≤b(A)limN→∞(∥∥χ1f∥∥+N∑n=2n(n−1)−α∥∥(χn−χn−1)f∥∥).

(39)

Now using that ∑n∥(χn−χn−1)f∥2=∥f∥2 and applying the
Cauchy-Schwartz inequality we get from Eq. (39)

Iv Proof of the Main Theorem

We shall need an analogue of the IMS localization formula, see Ref. ims, . The
functions Ja∈C∞(R3N−3) form the partition
of unity ∑aJ2a=1 and are homogeneous of
degree zero in the exterior of the unit sphere, i.e.Ja(λx)=Ja(x) for λ≥1, |x|=1 (this makes |∇Ja| fall off at infinity). Additionally, there exists a constant C>0 such
that

suppJa∩{x||x|>1}⊂{x||xi−xj|≥C|x|%for$i∈Ca1,j∈Ca2$}.

(41)

The functions of the IMS decomposition can be chosen
sigalsays1 (); sigalsays2 ()
invariant under permutations of particle coordinates both in
Ca1 and in Ca2, hence [Ja,P(a)]=0. Note also that for all f∈H2(R3N−3) one has Jaf∈H2(R3N−3), c. f. Lemma 7.4 in Ref. liebloss,
(the proof in Ref. liebloss, easily extends to the case of
Sobolev spaces of higher order).
The following version of the IMS localization formula can be verified by the
direct substitution

The Hamiltonian Hab defined for a≠b contains intercluster
interactions of the following
four clusters Ca1∩Cb1, Ca1∩Cb2, Ca2∩Cb1, Ca2∩Cb2,
while all interaction cross–terms between these four clusters are contained in
Iab. (For
some partitions it might happen that one of the four clusters is empty). If we
define by P(ab)sp the projection operator on the proper
symmetry subspace for particles within the cluster Cas∩Cbp then by the HVZ theorem reed (); teschl (); beattie ()

infσ(Hab(λ)P(ab))≥Ethr(λ),

(49)

where we define

P(ab):=P(ab)11P(ab)12P(ab)21P(ab)22.

(50)

Note that [JaJb,P(ab)]=0.

Lemma 5.

Suppose that H(λ) satisfies R1−R4. If ψn\lx@stackrelw→ϕ0 then

limn→∞∥∥[1−P(a)thr(λn)]J2a(ψn−ϕ0)∥∥=0(a=1,…,N),

(51)

limn→∞∥∥J2a(ψn−ϕ0)∥∥=0(a=N+1,…,2N−1−1).

(52)

Proof.

Note that ϕ0∈D(H0) by Lemmas 1, 2(a) in Ref. 1, . For every g∈L2(R3N−3;Cns) we have

(g,[1−P]ϕ0)=([1−P]g,ϕ0)

=limn→∞([1−P]g,ψn)=limn→∞(g,[1−P]ψn)=0.

(53)

Because g in (53) is arbitrary we conclude that Pϕ0=ϕ0. Consequently, P(a)ϕ0=ϕ0.